Derivatives of Trig Functions Section 3.3

Derivatives of Sine and Cosine

Derivative Rules 𝑑 𝑑𝑥 sin 𝑥 = cos 𝑥 𝑑 𝑑𝑥 cos 𝑥 =− sin 𝑥

Come up with the following rules… 𝑑 𝑑𝑥 tan 𝑥 = 𝑑 𝑑𝑥 csc 𝑥 = 𝑑 𝑑𝑥 sec 𝑥 =

Find the derivative… 𝑦=2 sin 𝑥 −4 cos 𝑥 + tan 𝑥 𝑦= 𝑥 3 sec 𝑥 𝑦=𝑥 𝑒 𝑥 cot 𝑥

Application Find an equation of the tangent line to the curve 𝑦=3𝑥+6 cos 𝑥 at the point where 𝑥= 𝜋 3 . Graph the curve and the tangent line on the same coordinate plane.

Application For what values of 𝑥 does the graph of 𝑓 have a horizontal tangent? 𝑓 𝑥 = 𝑒 𝑥 cos 𝑥

Application A mass on a spring vibrates horizontally on a smooth level surface. Its equation of motion is 𝑥 𝑡 = 8 sin 𝑡 , where 𝑡 is in seconds and 𝑥 is in centimeters. Find the velocity and the acceleration at time 𝑡. Find the position, velocity, and acceleration of the mass at time 𝑡= 2𝜋 3 . In what direction is it moving at that time?